A curvature-based approach for left ventricular shape analysis from cardiac magnetic resonance imaging

  • Si Yong Yeo
  • Liang Zhong
  • Yi Su
  • Ru San Tan
  • Dhanjoo N. Ghista
Original Article


It is believed that left ventricular (LV) regional shape is indicative of LV regional function, and cardiac pathologies are often associated with regional alterations in ventricular shape. In this article, we present a set of procedures for evaluating regional LV surface shape from anatomically accurate models reconstructed from cardiac magnetic resonance (MR) images. LV surface curvatures are computed using local surface fitting method, which enables us to assess regional LV shape and its variation. Comparisons are made between normal and diseased hearts. It is illustrated that LV surface curvatures at different regions of the normal heart are higher than those of the diseased heart. Also, the normal heart experiences a larger change in regional curvedness during contraction than the diseased heart. It is believed that with a wide range of dataset being evaluated, this approach will provide a new and efficient way of quantifying LV regional function.


Left ventricular shape Surface curvature Local surface fitting MRI Regional function 

1 Introduction

The shape of the left ventricle (LV) has intrigued physiologists attempting to gain a better understanding of its mode of operation, as well as clinicians trying to obtain diagnostic information on its performance [14, 25]. In general, the LV responds to volume overload by dilating to a more spherical shape and to pressure overload with hypertrophy in patients with ischemic cardiomyopathy. Furthermore, surgical procedures (e.g., the Dor procedure) can enable restoration of LV shape [15]. Accordingly, the shape of the LV is an important diagnostic and therapeutic index for evaluating a variety of cardiac disease states. Furthermore, knowledge of the LV remodeling process is important to (1) better understand and track the LV response to pressure and volume overload, (2) elucidate the mechanism of LV dysfunction in cardiac conditions that affect the chamber, and (3) better correlate patient symptoms to the underlying pathophysiology. In view of the association of LV shape with cardiac disease states and LV dysfunction, we aimed to develop a method for three-dimensional shape analysis of the LV by combining MRI and computational methods.

In the previous studies, the shape analysis methods were mainly on two-dimensional (2D) tomographic section (eg, 2D echocardiography and magnetic resonance imaging) of the heart using simple indices [12, 38] or three-dimensional (3D) echocardiography using sophistical curvature analysis [17]. There are, however, various shortcomings in using echocardiography [31], ventriculography [22], angiography [33] and indicator-dilution methods [13] compared with magnetic resonance imaging (MRI).

In recent years, computerized image segmentation has played an increasingly important role in the field of medical imaging, and has been applied in various applications [2, 4, 11]. Some popular methods used for medical image segmentation include thresholding [30], edge-based methods [26], deformable models [24] and level-sets methods [6]. Although these methods provide an automatic approach to image segmentation, they often produce problems such as “leaking” through weak and diffuse edges, and over-segmentation due to the excessive number of local minima when proper “denoising” is not provided [6, 26].

In fact, because of the relatively large spacing between the MR image slices (typically 5–10 mm for cardiac MRI in clinical practices) and the inhomogeneous contrast of the myocardium, these segmentation methods may not be suitable for the delineation of the LV contours [6, 26]. Also, existing automatic segmentation methods for CMR images are often based on algorithms that are sensitive to image quality and frequently depend on the specific imaging protocol [1, 10, 19, 29]. Therefore, in this work, the segmentation of LV endocardial contours has been performed manually by a trained expert using CMRtools, which is a software package for the display and quantitative analysis of cardiac MR images [32].

Existing methods for estimating higher order properties, such as curvatures, of discrete surface information (e.g., triangulated mesh) can be classified into three general categories. The first group of methods computes the curvature directly from the triangulation by using the concept of discrete differential geometry [5, 18]. Although these methods involve low computational cost, their accuracies are largely affected by the quality of the triangulation, and they often become unstable near degenerate configurations. The second group of methods uses an analytical approach to extract the differential properties and curvature information by means of local surface fitting [36, 37, 39]. In general, a quadratic polynomial function is used to approximate a smooth surface to a local set of points in a region. These methods are generally robust and produce accurate results as shown by Garimella and Swartz [21], and Cazals and Pouget [7]. The third group of methods extracts curvature information by computing the average of the curvature tensor over a small area of the triangulated surface [3, 35]. These tensor-based methods often produce significant errors in estimating curvature, even for densely triangulated models and simple geometries such as a sphere or ellipsoid.

In this paper, we have chosen to characterize the LV shape by deriving the LV surface curvatures via an analytical approach using a surface patch fitting method, due to its robustness and accuracy. We have applied this method to five normal subjects and five heart disease patients.

2 Methods

2.1 MRI scans

This study involved five normal subjects and five patients with heart diseases. All subjects underwent diagnostic MRI scan. None of the normal subjects had (1) significant valvular or congenital cardiac disease, (2) history of myocardial infarction, (3) coronary artery lesions, or (4) abnormal left ventricular pressure, end-diastolic volume or ejection fraction.

Cine MR images of the LV were acquired on a 1.5 T MR scanner (Avanto, Simens Medical Solutions, Erlangen) at the National Heart Centre, Singapore. Preliminary short-axis acquisition located the plane passing through the mitral and aortic valves. This allowed the location and acquisition of the oblique long-axis plane of the LV, which is orthogonal to the short-axis plane and passes through the mitral valve, apex and aortic valve. In addition, vertical long-axis planes were acquired. True (fast imaging with steady-state precessing) MR pulse sequence with segmented k-space and retrospective electrocardiographic gating was used to acquire a parallel stack of 2D cine images of the LV in the short-axis plane from the LV base to apex (8 mm interslice thickness, no interslice gap). The field of view was typically 320 mm with in-plane spatial resolution of less than 1.5 mm. Each slice was acquired in a single breath hold with 25 temporal phases per heart cycle. Figure 1 depicts some images of the short-axis plane and the oblique long-axis plane as seen from below. The entire image acquisition duration averaged 30 mins.
Fig. 1

Example segmented TrueFISP 2D cine MR images of short axis slices acquired at the base, middle and apex (from top to bottom, respectively) of the left ventricle. The end-diastolic and end-systolic phases are depicted in the left and right columns, respectively (a). The two- and four-chamber oblique long-axis planes are depicted in (b)

2.2 Data processing and geometry reconstruction

Due to the relatively large spacing between the cardiac MR image slices as well as the contrast inhomogeneity of the myocardial tissue, it is not efficient to segment the LV contours using automatic methods. Instead, the segmentation of LV endocardial contours is performed manually by a trained expert using CMRtools.

Both short- and long-axis images were used in this study, so as to increase the accuracy and interactivity of the segmentation process. For each time frame, control points are placed on the endocardial contours by the user, such that they are constrained to lie on the intersections of the short- and long-axis image views (Fig. 2a–d). In order to accurately define the LV geometry, long-axis contour planes were further reconstructed by interactively fitting B-spline curves to the short-axis images (Fig. 2e, f). In so doing, a radial set of long-axis contour planes which are orthogonal to the short-axis images are created; this provides a more comprehensive set of contours points for defining the LV geometry. The set of LV contour points derived from the segmentation process are then triangulated using our in-house toolkit to produce the reconstructed 3D LV surface.
Fig. 2

Top Adding of control points on the endocardial contours at a short- and b long-axis image plane at end-diastole (ED) phase. Middle Corresponding endocardial contours at c short- and d long-axis image at end-systole (ES) phase. Bottom 3D LV depicting long- and short-axis image planes contours, created at e ED phase and f ES phase using B-spline curves

2.3 Computation of three-dimensional shape descriptor

In this paper, the intrinsic LV surface properties are computed via an analytical approach, using a surface patch fitting method. Such surface fitting methods have been used by researchers to investigate the surface curvatures of anatomical structures of the human body, and also to define anatomical landmarks for various applications [16, 28].

2.3.1 Local surface patch fitting

In the vicinity of a point x, the surface can be approximated by an osculating paraboloid which may be represented by a quadratic polynomial with parameters du and dv. This polynomial is represented as a Taylor expansion at x of the paraboloid surface, by omitting the higher order terms after the quadratic term [20],
$$ {\bf x} \left( {u + {\text{d}}u,v + {\text{d}}v} \right) = {\bf x}_{00} + {\bf x}_{u} \,{\text{d}}u + {\bf x}_{v} \,{\text{d}}v + \left( {{\bf x}_{uu} \,{\text{d}}u^{2} + 2{\bf x}_{uv} \,{\text{d}}u\,{\text{d}}v + {\bf x}_{vv} \,{\text{d}}v^{2} } \right)/2 $$
The coefficients x00, xu, xv, xuu, xuv and xvv are the zero, first and second derivatives of x with respect to u and v at the surface point x00 = x(u, v). With these six derivatives, the curvatures of the surface can be computed. A function given by z = f(x, y), which is a second order polynomial of the form
$$ z = c_{1} + c_{2} p + c_{3} q + c_{4} p^{2} + c_{5} pq + c_{6} q^{2} $$
is used to fit the approximating surface patch. In Eq. 2, p = x – x0 and q = y – y0, where x0 and y0 are the x and y coordinates of the center point x0 of the surface patch under consideration. The six coefficients of the paraboloid are then obtained by the least-squares solution of an over-determined system of linear equations
$$ \left\{ {\begin{array}{*{20}c} {z_{1}^{'} } \\ {z_{2}^{'} } \\ {z_{3}^{'} } \\ \vdots \\ {z_{n}^{'} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} 1 & {p_{1} } & {q_{1} } & {p_{1}^{2} } & {p_{1} q_{1} } & {q_{1}^{2} } \\ 1 & {p_{2} } & {q_{2} } & {p_{2}^{2} } & {p_{2} q_{2} } & {q_{2}^{2} } \\ 1 & {p_{3} } & {q_{3} } & {p_{3}^{2} } & {p_{3} q_{3} } & {q_{3}^{2} } \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & {p_{n} } & {q_{n} } & {p_{n}^{2} } & {p_{n} q_{n} } & {q_{n}^{2} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {c_{1} } \\ {c_{2} } \\ {c_{3} } \\ {c_{4} } \\ {c_{5} } \\ {c_{6} } \\ \end{array} } \right\} $$
where (1) n is the number of points in the neighborhood of a selected surface point, and (2) the coefficients Ci are given by Eq. 2.
It is desired that the coefficients c1c6 are acquired such that the divergence of the fitted paraboloid and the data points are minimized, which is similar to minimizing the differences between zn (measured value of z) and z′n (calculated value of z). The minimization problem at this point is then expressed in terms of error function E, as
$$ E(z) = \sum\limits_{n = 1}^{N} {G_{n} (z_{n}^{'} - z_{n} )^{2} } = \sum\limits_{n = 1}^{N} {G_{n} (z(p_{n} ,q_{n} ) - z_{n} )^{2} } $$
where N represents the number of points within the fitted paraboloid, and G is a distance weighting function, such that
$$ G_{i} = f (e^{{( - (p_{{_{i} }}^{2} + q_{i}^{2} )/d^{2} )}} ),\quad i = 1,2,3, \ldots, N $$
where f and d are arbitrary constants which can be adjusted accordingly. The error function E will have a minimum when
$$ \frac{\partial E}{{\partial c_{i} }} = 0,\quad i = 1, 2, 3, 4, 5, 6 $$
thus yielding six linear equations required to compute the six coefficients.
The first coefficient c1 is actually the z value of the center point x0 on the fitted surface patch, while the other coefficients c2c6 can be interpreted as the first and second derivatives of x with respect to p and q at x0. With these coefficients, we can easily derive the principal curvatures κ1 and κ2, using the theory of differential geometry [9, 34]. The shape descriptor, which is based on the curvedness C (at a point x) as presented by Koenderink [27], can then be defined as
$$ C = \sqrt {\frac{{\kappa_{1}^{2} + \kappa_{2}^{2} }}{2}} $$

The index C is a positive value, and describes the magnitude of the curvedness at a point, and thus is a measure of how gently or highly curved a surface is.

2.3.2 Computation of LV surface curvatures

As shown in Fig. 3, the procedure for determining the LV surface curvatures from cardiac MR images can be described as follows:
Fig. 3

Procedure for acquiring LV surface curvatures from cardiac MR images

  1. 1.

    Interactively segment the LV contours from the cardiac MR images.

  2. 2.

    Reconstruct the LV surface by triangulation of the LV contour points derived from the segmentation process.

  3. 3.

    For each vertex, select the n-ring neighboring vertices.

  4. 4.

    At each selected vertex, perform quadratic surface patch fitting and compute the surface curvature.


The choice of the n-ring neighboring points used for the patch fitting is arbitrary. However as shown in the previous section, there are six unknowns c1c6 in the quadratic function. Therefore, at least five neighboring points around the selected vertex are required for the local patch fitting, and a 1-ring neighborhood may produce errors in regions with insufficient points. On the other hand, selecting a region with too many points (eg, a 5-ring neighborhood) will cause an “over-smoothing” of the surface patch being considered. Therefore in this work, local surface fitting and curvature computation at each vertex is performed based on a set of 2-ring neighboring points.

In order to perform regional analysis of the LV surface shape, the reconstructed LV surface is divided into 16 segments (Fig. 4) using the standardized myocardial segmentation and nomenclature described in [8]. In particular, the LV endocardial surface is divided into three different segments (i.e., basal, mid and apical). The point separating the basal anterior and anteroseptal region is identified from the original MR image, using the intersection of the right ventricular wall and the LV as a reference point. The basal and mid segments are then divided into six equal regions, and the apical segment is divided into four regions. The regional (or mean) curvedness CRgn for each segment is then defined (in terms of the curvedness Ci of the vertices of the segment) as:
$$ C_{\text{Rgn}} = \frac{1}{N}\sum\limits_{i}^{N} {C_{i} } ,\quad i = 1, 2, 3, \ldots, N $$
where N denotes the number of vertices within each region and Ci is given by Eq. 7.
Fig. 4

Standardized myocardial segmentation and nomenclature [8]. 1 basal anterior, 2 basal anteroseptal, 3 basal inferoseptal, 4 basal inferior, 5 basal inferolateral, 6 basal anterolateral, 7 mid anterior, 8 mid anteroseptal, 9 mid inferoseptal, 10 mid inferior, 11 mid inferolateral, 12 mid anterolateral, 13 apical anterior, 14 apical septal, 15 apical inferior and 16 apical lateral

We then define a normalized index to measure the extent of surface curvedness change at each LV region from end-diastolic (ED) to end-systolic (ES) phases, known as the percent regional curvedness phase-change ΔCRgn, such that
$$ \Updelta C_{\text{Rgn}} = \frac{{C_{\text{Rgn,ES}} - C_{\text{Rgn,ED}} }}{{C_{\text{Rgn,ES}} }} \times 100\% $$
where CRgn,ED and CRgn,ES are regional curvedness computed at ED and ES, respectively. A positive ΔCRgn indicates that a region becomes more curved from ED to ES, and a negative ΔCRgn indicates that a region becomes flatter during the systolic phase.

3 Results

We have created a software package, Cardiac Shape Analysis tools (CSAtools), which can be used to efficiently process the LV geometrical data for shape analysis. This software tool is designed for the reconstruction of cardiac surface models from input contour points and for the analysis of cardiac regional shape, to compute curvedness at a point, regional curvedness, and percent change in regional curvedness from ED to ES. For example, the plots of LV surface curvedness (C) at ED and ES, for both a normal and a diseased heart, are depicted in Fig. 5. For these subjects, Fig. 6 depicts the regional curvedness CRgn (mm−1) in the 16-segment heart model at ED and ES, and the corresponding percent regional curvedness change (ΔCRgn). It is shown that the CRgn values for each region of the normal heart are relatively higher than those found in the corresponding region of the diseased heart at both ED and ES. For the normal heart, the apical region is found to experience the largest curvedness change from ED to ES (i.e., ΔCRgn = 55% at apical anterior, ΔCRgn = 62% at apical septal, ΔCRgn = 56% at apical inferior, ΔCRgn = 67% at apical lateral), followed by the mid ventricle region with ΔCRgn ranging from 42 to 61%. A considerable change in regional curvedness can also be observed at the basal region (i.e., 16% ≤ ΔCRgn ≤ 45%). The basal region, in general, has the smallest curvedness change, because the LV is anchored to the aorta at the base, while the apex has maximal laxity and undergoes maximal twisting motion and hence strain deformation.
Fig. 5

LV surface curvedness (C) plots at end-diastolic (ED) and end-systolic (ES) phase for a normal and b diseased heart. Note that the surface models are truncated at the apex region for visualization purposes

Fig. 6

Regional plots of 16-segments heart model showing LV regional curvedness, CRgn (mm−1) at end-diastolic (ED) and end-systolic (ES) phases, and the associated percent regional curvedness phase-change (ΔCRgn), evaluated for a normal and b diseased heart subjects, respectively. Note that CRgn,ED, CRgn,ES and ΔCRgn values computed at each region of the normal heart are relatively higher than those at the corresponding region of the diseased heart at both ED and ES phases. For the normal heart, ΔCRgn values evaluated at the apical and mid ventricle regions are relatively higher than those found at the basal region

Table 1 summarizes the hemodynamic and volumetric parameters of ten subjects. In heart disease patients, the LV ejection fraction (LVEF 65 ± 5 vs. 18 ± 4%; p < 0.05) was significantly lower than those in normal subjects, and their LV end-diastolic index (146 ± 20 vs. 68 ± 7 ml/m2; p < 0.05) and end-systolic volume index (119 ± 15 vs. 24 ± 5 ml/m2; p < 0.05) were greater than those in normal subjects. The computed values of curvedness at ED (CRgn,ED) and ES (CRgn,ES) and their corresponding percent curvedness phase-change (∆CRgn) in each segment, for each subject, are summarized in Table 2.
Table 1

Characteristics of normal and heart failure subjects


Normal (= 5)

Heart disease (= 5)

p value

Age (years)

34 ± 24

51 ± 10


Weight (kg)

71 ± 20

68 ± 18


Height (cm)

169 ± 10

160 ± 7


Diastolic blood pressure (mmHg)

74 ± 9

72 ± 14


Systolic blood pressure (mmHg)

133 ± 11

108 ± 8


HR (beats/min)

70 ± 8

75 ± 19


CI (ml/m2)

3.3 ± 0.2

2.1 ± 0.2


EDVI (ml/m2)

68 ± 7

146 ± 20


ESVI (ml/m2)

24 ± 5

119 ± 15


EF (%)

65 ± 5

18 ± 4


Table 2

LV regional curvedness, CRgn (mm−1) at end-diastolic (ED) and end-systolic (ES) phases, and the associated percent regional curvedness phase-change (ΔCRgn) in the 16-segment heart model for normal and diseased heart subjects


Normal (n = 5)

Diseased (n = 5)

CRgn,ED (mm−1)

CRgn,ES (mm−1)

CRgn (%)

CRgn,ED (mm−1)

CRgn,ES (mm−1)

CRgn (%)

Basal anterior

0.045 ± 0.0052

0.066 ± 0.0052

32 ± 9

0.044 ± 0.025

0.048 ± 0.024

9 ± 9*

Basal anteroseptal

0.044 ± 0.0070

0.065 ± 0.0048

32 ± 9

0.040 ± 0.013

0.046 ± 0.016

12 ± 8*

Basal inferoseptal

0.037 ± 0.0061

0.054 ± 0.0066

33 ± 10

0.041 ± 0.018

0.041 ± 0.018

2 ± 18*

Basal inferior

0.047 ± 0.0074

0.063 ± 0.0056

26 ± 8

0.043 ± 0.018

0.043 ± 0.022

−3 ± 26*

Basal inferolateral

0.036 ± 0.0066

0.056 ± 0.0081

35 ± 3

0.038 ± 0.019

0.041 ± 0.021

7 ± 21*

Basal anterolateral

0.036 ± 0.0048

0.056 ± 0.0063

34 ± 15

0.044 ± 0.026

0.041 ± 0.025

−8 ± 17*

Mid anterior

0.051 ± 0.016

0.075 ± 0.014

37 ± 13

0.041 ± 0.017

0.045 ± 0.013*

15 ± 12*

Mid anteroseptal

0.054 ± 0.015

0.091 ± 0.024

40 ± 14

0.037 ± 0.007

0.042 ± 0.004*

15 ± 17*

Mid inferoseptal

0.045 ± 0.0099

0.081 ± 0.0140

44 ± 14

0.046 ± 0.018

0.049 ± 0.012*

9 ± 25*

Mid inferior

0.057 ± 0.016

0.084 ± 0.015

31 ± 17

0.046 ± 0.017

0.053 ± 0.013*

12 ± 30*

Mid inferolateral

0.047 ± 0.0093

0.073 ± 0.020

32 ± 22

0.039 ± 0.012

0.042 ± 0.007*

8 ± 15*

Mid anterolateral

0.044 ± 0.0010

0.065 ± 0.020

34 ± 16

0.044 ± 0.017

0.041 ± 0.021

−15 ± 21*

Apical anterior

0.087 ± 0.0145

0.215 ± 0.093

56 ± 11

0.049 ± 0.015*

0.053 ± 0.010*

8 ± 15*

Apical septal

0.110 ± 0.042

0.209 ± 0.082

48 ± 13

0.044 ± 0.008*

0.049 ± 0.008*

10 ± 9*

Apical inferior

0.124 ± 0.044

0.243 ± 0.085

48 ± 14

0.058 ± 0.016*

0.062 ± 0.011*

13 ± 6*

Apical lateral

0.108 ± 0.039

0.272 ± 0.127

56 ± 20

0.056 ± 0.010*

0.059 ± 0.010*

10 ± 16*

Data are means ± SD

* Significant difference compared with data in normal subjects (p < 0.05)

It is found that ∆CRgn for different regions of the normal heart are much higher than those found for corresponding regions of the diseased heart. This relatively large difference in ΔCRgn is due to the large surface deformation (due to good LV contraction during systole) of the normal heart, as compared to the minimally deformed diseased heart (due to impaired LV function) during contraction. For example, the mean values of ΔCRgn at the apical region of the diseased heart are found to be 8, 10, 13 and 10% at the apical anterior, septal, inferior and lateral segments, respectively, which are lower than those observed for the normal heart. Note the negative ΔCRgn (i.e., −15%) found at the mid anterolateral segment of the diseased heart, this indicates that instead of becoming more curved during the systolic phase, this region of the LV becomes flatter. These lower and negative ΔCRgn values found in the diseased heart may be attributed to the impaired LV systolic function and regional contractile function of the infarcted myocardial segments.

4 Discussion

In this paper, the differential properties and curvature information of the LV surface are extracted via an analytical approach by means of local surface fitting [36, 39]. This approach is robust and produces accurate results, as shown by Garimella and Swartz [21], and Cazals and Pouget [7]. Here, we examine the LV shape derived from anatomically accurate reconstructed 3D models of the LV, and are able to assess LV regional shape in terms of curvatures. This is in contrast to previous studies, which limited the analysis to simplified shape indices or parameters and 2D models based on LV short- and long-axes [23, 38, 40].

The Gaussian (K) and mean curvatures (H) are considered the most widely used indicators for surface shape classification. Given a point on a surface, K is used to categorize the point into elliptic, hyperbolic, parabolic or planar based on the sign of K, while H can be used to differentiate concave (i.e., −ve H) regions from convex (i.e., +ve H) regions. However, Koenderink and van Doorn [27] have shown that K and H are not very indicative of local shape, and introduced more significant measures of local shape known as the shape index (S) and curvedness (C). In their formulation, S is scale invariant and provides a smooth categorization of shapes, such as concave shapes (−1 < S < − 1/2), hyperboloid shapes (−1/2 < S < 1/2) and convex shapes (1/2 < S < 1). The curvedness C describes the magnitude of the curvature at a surface point. It is a measure of how highly or gently curved a point is.

Since we are interested in the magnitude or extent of curvature changes (i.e., surface deformation as the LV contracts) and not the categorization of shapes or directions of curvatures, we chose to use C as the curvature metric for analyzing LV regional function. This metric has an advantage over H, in the situation when the mean curvature value vanishes at surface points where κ1 = −κ2 and its magnitude is not intuitive (i.e., H is small although surface is highly curved with κ1 ≈ −κ2, since H is an average of the principal curvatures). Also, at surface regions where the H values between two time frames have the same magnitude but differ in sign convention, ∆H gives a value of zero even though a significant deformation is involved.

In this study, we are able to elucidate and assess the LV surface shape and its deformation in terms of curvatures. The results show that the normal LV has significantly higher surface curvedness (i.e., high C value) as compared to the diseased heart (i.e., low C value) at the end-systolic phase. This is due to the relatively greater curved surface of the normal LV chamber in contrast to the dilated LV chamber of the diseased heart. We have also demonstrated that the normal LV experiences a significant increase in curvedness (i.e., high ΔCRgn) when it contracts during systole, while the diseased LV undergoes minimal change in curvedness (i.e., low ΔCRgn) during systole. This considerable large difference in ΔCRgn between the normal and diseased hearts illustrates that the normal LV contracts and deforms more than the diseased LV. This finding also provides testimony to the sensitivity and accuracy of our 3D LV shape reconstruction methodology and our application of the surface curvedness indicators.

The CSAtools can be used to read in segmented contour points, to reconstruct the LV surface, compute and visualize LV curvature distribution, and analyze the regional LV curvatures. This set of procedures (i.e., from input contour points to output regional curvatures) takes only a few minutes to perform, and thus provides an efficient and convenient tool for clinical diagnosis. This is in contrast to methods using finite element simulation of LV wall deformation and myocardial stress/strain analysis, which usually take several hours or even days for the simulation to converge. In addition, the percent curvedness phase-change (∆C) plots in Table 2 enable us to assess LV regional function easily, which may be a valuable tool for detecting unusual LV wall activities as well as ischemic and infarcted myocardial segments.

There are many clinical implications based on the assessment of LV shape indices. By studying the surface curvedness of the LV endocardial surface during the cardiac cycle, one can get a better perception of the regional wall deformation of the LV. Also, the knowledge of normal values of regional curvedness and percent curvedness change in the LV may allow the identification and follow-up of local functional abnormities. We envisage that with a wide range of LV datasets being tested and evaluated, we can easily correlate the changes in LV surface curvedness with the intricate LV regional function. This will eventually be useful for the assessment of regional cardiac function, especially for the detection of infarct regions and unusual wall activities.

4.1 Limitations

The accuracy of this method depends on factors such as image resolution and the surface reconstruction process. The spacing between short-axis image slices for cardiac MRI in clinical practices is typically around 5–10 mm. Therefore, a considerable amount of interpolation between image slices has been used, and this may affect the accuracy of the curvature evaluation. The accuracy of the surface curvatures also depends on the mesh quality and densities of the LV models, since the differential properties of each surface point are computed using the neighboring vertices. Due to the resolution of the MR images used and a relatively more intricate shape of the apical region, segmentation of LV contours is often more difficult in the apical region than in the mid and basal regions. This could affect the evaluation of the regional curvedness at the apical region, especially at ES. In this study, five healthy LVs and five diseased LVs have been evaluated. A larger range of dataset needs to be assessed so that one can get a more accurate correlation between LV curvature changes and LV regional function (i.e., ventricular wall motion). Currently, more work is being done in this direction.

5 Conclusions

We have developed a framework for the rapid and accurate assessment of LV shape and its associated changes during the cardiac cycle. The set of procedures pertain to (1) the segmentation and reconstruction of anatomically accurate LV models from cardiac MR images, and (2) the extraction of differential properties of the LV surface via analytic surface fitting technique. This enables us to analyze LV regional shape and deformation efficiently and accurately, in terms of surface curvedness. The regional curvedness of different LV segments and their associated percent curvedness phase-changes are evaluated for normal and diseased hearts, using the above-mentioned methodology. It is shown that the regional surface curvedness and their changes (from ED to ES) for the normal heart differ greatly from those of the diseased heart. Thus, this new approach may be an efficient way of quantifying LV regional function.



This work was supported in part by a research grant from the National Heart Centre, Singapore.


  1. 1.
    Angelie E, de Koning PJ, Danilouchkine MG, van Assen HG, Koning G, van der Geest RJ, Reiber JK (2005) Optimizing the automatic segmentation of the left ventricle in magnetic resonance images. Med Phys 32(2):369–375CrossRefGoogle Scholar
  2. 2.
    Al-Attar SA, Pollex RL, Robinson JF, Miskie BA, Walcarius R, Rutt BK, Hegele RA (2006) Semi-automated segmentation and quantification of adipose tissue in calf and thigh by MRI: a preliminary study in patients with monogenic metabolic syndrome. BMC Med Imaging 6:11CrossRefGoogle Scholar
  3. 3.
    Alliez P, Cohen-Steiner D, Devillers O, Levy B, Desbrun M (2003) Anisotropic polygonal remeshing. ACM Trans Graphics 22(3):485–493CrossRefGoogle Scholar
  4. 4.
    Ayres FJ, Zuffo MK, Rangayyan RM, Boag GS, Odone Filho V, Valente M (2004) Estimation of the tissue composition of the tumor mass in neuroblastoma using segmented CT images. Med Biol Eng Comput 42(3):366–377CrossRefGoogle Scholar
  5. 5.
    Borrelli V, Cazals F, Morvan J-M (2003) On the angular defect of triangulations and the pointwise approximation of curvatures. Comput Aided Geom Design 20(6):319–341MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Brox T, Weickert J (2006) Level set segmentation with multiple regions. IEEE Trans Image Process 15(10):3213–3218CrossRefGoogle Scholar
  7. 7.
    Cazals F, Pouget M (2003) Estimating differential quantities using polynomial fitting of osculating jets. In: Proceedings of symposium on geometry processing, pp 177–187Google Scholar
  8. 8.
    Cerqueira MD, Weissman NJ, Dilsizian V, Jacobs AK, Kaul S, Laskey WK, Pennell DJ, Rumberger JA, Ryan T, Verani MS (2002) Standardized myocardial segmentation and nomenclature for tomographic imaging of the heart: a statement for healthcare professionals from the Cardiac Imaging Committee of the Council on Clinical Cardiology of the American Heart Association. Circulation 105(4):539–542CrossRefGoogle Scholar
  9. 9.
    Chuiv NN (1985) Differential geometry for surveying engineers. University of New Brunswick, FrederictionGoogle Scholar
  10. 10.
    Corsi C, Vernesi F, Lamberti C, Mor-Avi V (2006) Improved automated quantification of left ventricular size and function from cardiac magnetic resonance images. Comput Cardiol 33:53–56Google Scholar
  11. 11.
    Cotin S, Delingette H, Ayache N (2000) A hybrid elastic model allowing real-time cutting, deformations and force-feedback for surgery training and simulation. Vis Comput 16(8):437–452MATHCrossRefGoogle Scholar
  12. 12.
    D’Cruz IA, Aboulatta H, Killam H, Bradley A, Hand RC (1989) Quantitative two-dimensional echocardiographic assessment of left ventricular shape in ischemic heart disease. J Clin Ultrasound 17(8):569–572CrossRefGoogle Scholar
  13. 13.
    Debatin J, Nadel SN, Sostman HD, Spritzer CE, Evans AJ, Grist TM (1992) Magnetic resonance imaging-cardiac ejection fraction measurements Phantom study comparing four different methods. Invest Radiol 27(3):198–204CrossRefGoogle Scholar
  14. 14.
    Devereux RB (1995) Left ventricular geometry, pathophysiology and prognosis. J Am Coll Cardiol 25(4):885–887CrossRefGoogle Scholar
  15. 15.
    Dor V (2001) The endoventricular circular patch plasty (“Dor” procedure) in ischemic akinetic dilated ventricles. Heart Fail Rev 6(3):187–193CrossRefGoogle Scholar
  16. 16.
    Drerup B, Hierholzer E (1985) Objective determination of anatomical landmarks on the body surface: measurement of the vertebra prominens from surface curvature. J Biomech 18(6):467–474CrossRefGoogle Scholar
  17. 17.
    Duann JR, Chiang SH, Lin SB, Lin CC, Chen JH, Su JL (1999) Assessment of left ventricular cardiac shape by the use of volumetric curvature analysis from 3D echocardiography. Comput Med Imag Graph 23(2):89–101CrossRefGoogle Scholar
  18. 18.
    Dyn N, Hormann K, Kim SJ, Levin D (2000) Optimizing 3D triangulations using discrete curvature analysis. In: Lyche T, Shumaker LL (eds) Mathematical methods for curves and surfaces. Vanderbilt University Press, Nashville, pp 135–146Google Scholar
  19. 19.
    El Berbari R, Bloch I, Redheuil A, Angelini E, Mousseaux E, Frouin F, Herment A (2007) An automated myocardial segmentation in cardiac MRI. In: Proceedings of the 29th annual international conference of the IEEE EMBS, pp 4508–4511Google Scholar
  20. 20.
    Frobin W, Hierholzer E (1982) Analysis of human back shape using surface curvatures. J Biomech 15(5):379–390CrossRefGoogle Scholar
  21. 21.
    Garimella RV, Swartz BK (2003) Curvature estimation for unstructured triangulations of surfaces. Los Alamos National LaboratoryGoogle Scholar
  22. 22.
    Gaudio C, Tanzilli G, Mazzarotto P, Motolese M, Romeo F, Marino B, Reale A (1991) Comparison of left ventricular ejection fraction by magnetic resonance imaging and radionuclide ventriculography in idiopathic dilated cardiomyopathy. Am J Cardiol 67(5):411–415CrossRefGoogle Scholar
  23. 23.
    Halmann M, Sideman S, Azhari H, Markiewitz W, Beyar R (1991) Dynamic analysis of left ventricular shape based on curvature function. Basic Res Cardiol 86(4):393–401CrossRefGoogle Scholar
  24. 24.
    Han X, Xu C, Prince JL (2002) A topology preserving geometric deformable model and its application in brain cortical surface reconstruction. In: Osher S, Paragios N (eds) Geometric level set methods in imaging, vision and graphics. Springer, HeidelbergGoogle Scholar
  25. 25.
    Harjai KJ, Edupuganti R, Nunez E, Turgut T, Scott L, Pandian NG (2000) Does left ventricular shape influence clinical outcome in heart failure? Clin Cardiol 23(11):813–819CrossRefGoogle Scholar
  26. 26.
    Iannizzotto G, Vita L (2000) Fast and accurate edge-based segmentation with no contour smoothing in 2-D real images. IEEE Trans Image Process 9(7):1232–1237CrossRefGoogle Scholar
  27. 27.
    Koenderink JJ, van Doorn AJ (1992) Surface shape and curvature scales. Image Vis Comput 10(8):557–565CrossRefGoogle Scholar
  28. 28.
    Liu X, Kim W, Drerup B, Mahadev A (2005) Tibial torsion measurement by surface curvature. Clin Biomech 20(4):443–450CrossRefGoogle Scholar
  29. 29.
    Lynch M, Ghita O, Whelam PF (2006) Automatic segmentation of the left ventricle cavity and myocardium in MRI data. Comput Biol Med 36:389–407CrossRefGoogle Scholar
  30. 30.
    Mardia KV, Hainsworth TJ (1998) A spatial thresholding method for image segmentation. IEEE Trans Pattern Anal Mach Intell 10:919–927CrossRefGoogle Scholar
  31. 31.
    Pini R, Giannazzo P, Di Bari M, Innocenti F, Rega L, Casolo G, Devereux R (1997) Transthoracic three-dimensional echocardiography reconstruction of left and right ventricles: in vitro validation and comparison with magnetic resonance imaging. Am Heart J 133(2):221–229CrossRefGoogle Scholar
  32. 32.
    Saber NR, Wood NB, Gosman AD, Merrifield RD, Yang GZ, Charrier CL, Gatehouse PD, Firmin DN (2003) Progress towards patient-specific computational flow modeling of the left heart via combination of magnetic resonance imaging with computational fluid dynamics. Ann Biomed Eng 31(1):42–52CrossRefGoogle Scholar
  33. 33.
    Semelka R, Tomei E, Wagners S, Mayo J, Kondo C, Suzuki J, Caputo G, Higgins C (1990) Normal left ventricular dimensions and function: reproducibility of measurements with cine MR imaging. Radiology 174(3):763–768Google Scholar
  34. 34.
    Willmore TJ (1985) An introduction to differential geometry. Oxford University Press, DelhiGoogle Scholar
  35. 35.
    Wu ST, Batagelo HC (2007) Estimating curvatures and their derivatives on meshes of arbitrary topology from sampling directions. Vis Comput 23(9):803–812CrossRefGoogle Scholar
  36. 36.
    Yeo SY, Tan RS, Chai GB, Ghista DN (2005) Variation of left ventricular surface shape during the cardiac cycle. In: Proceedings of the 3rd IASTED international conference on biomechanics, BenidormGoogle Scholar
  37. 37.
    Yeo SY, Zhong L, Su Y, Tan RS, Ghista DN (2007) Analysis of left ventricular surface deformation during isovolumic contraction. In: Proceedings of the 29th annual international conference of the IEEE Engineering in Medicine and Biology Society, Lyon, pp 787–790Google Scholar
  38. 38.
    Zhong L, Ghista DN, Ng EY, Lim ST, Chua T, Lee CN (2006) Left ventricular shape-based contractility index. J Biomech 39(13):2397–2409CrossRefGoogle Scholar
  39. 39.
    Zhong L, Yeo SY, Su Y., Le TT, Tan RS, Ghista DN (2007) Regional assessment of left ventricular surface shape from magnetic resonance imaging. In: Proceedings of the 29th annual international conference of the IEEE Engineering in Medicine and Biology Society, Lyon, pp 884–887Google Scholar
  40. 40.
    Zhong L, Ghista DN, Ng EY, Lim ST, Chua TS, Lee CN (2007) Effect of left ventricular shape alteration on contractility and function. J Med Eng Technol 31(4):253–262CrossRefGoogle Scholar

Copyright information

© International Federation for Medical and Biological Engineering 2008

Authors and Affiliations

  • Si Yong Yeo
    • 1
  • Liang Zhong
    • 2
    • 3
  • Yi Su
    • 1
  • Ru San Tan
    • 2
  • Dhanjoo N. Ghista
    • 4
  1. 1.Institute of High Performance ComputingSingaporeSingapore
  2. 2.Department of CardiologyNational Heart CentreSingaporeSingapore
  3. 3.College of Life Science and TechnologyHuazhong University of Science and TechnologyWuhanPeople’s Republic of China
  4. 4.Parkway AcademySingaporeSingapore

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